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noetherNormalization -- data for Noether normalization

Synopsis

Description

The computations performed in the routine noetherNormalization use a random linear change of coordinates, hence one should expect the output to change each time the routine is executed.
i1 : R = QQ[x_1..x_4];
i2 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o2 : Ideal of R
i3 : (f,J,X) = noetherNormalization I

               7     3                   2                      13 2   3    
o3 = (map(R,R,{-x  + -x  + x , x , 5x  + -x  + x , x }), ideal (--x  + -x x 
               6 1   8 2    4   1    1   3 2    3   2            6 1   8 1 2
     ------------------------------------------------------------------------
                 35 3     191 2 2   1   3   7 2       3   2       2      
     + x x  + 1, --x x  + ---x x  + -x x  + -x x x  + -x x x  + 5x x x  +
        1 4       6 1 2    72 1 2   4 1 2   6 1 2 3   8 1 2 3     1 2 4  
     ------------------------------------------------------------------------
     2   2
     -x x x  + x x x x  + 1), {x , x })
     3 1 2 4    1 2 3 4         4   3

o3 : Sequence
The next example shows how when we use the lexicographical ordering, we can see the integrality of R/ f I over the polynomial ring in dim(R/I) variables:
i4 : R = QQ[x_1..x_5, MonomialOrder => Lex];
i5 : I = ideal(x_2*x_1-x_5^3, x_5*x_1^3);

o5 : Ideal of R
i6 : (f,J,X) = noetherNormalization I

               3                   2               1                    
o6 = (map(R,R,{-x  + 4x  + x , x , -x  + 3x  + x , -x  + 2x  + x , x }),
               2 1     2    5   1  3 1     2    4  2 1     2    3   2   
     ------------------------------------------------------------------------
            3 2                   3  27 3        2 2   27 2            3  
     ideal (-x  + 4x x  + x x  - x , --x x  + 27x x  + --x x x  + 72x x  +
            2 1     1 2    1 5    2   8 1 2      1 2    4 1 2 5      1 2  
     ------------------------------------------------------------------------
          2     9     2      4      3        2 2      3
     36x x x  + -x x x  + 64x  + 48x x  + 12x x  + x x ), {x , x , x })
        1 2 5   2 1 2 5      2      2 5      2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                          
     {-10} | 6x_1x_2x_5^6-864x_2^9x_5-6144x_2^9+108x_2^8x_5^2+1536x_2^8x_5-9x
     {-9}  | 3072x_1x_2^2x_5^3-54x_1x_2x_5^5+768x_1x_2x_5^4+7776x_2^9-972x_2^
     {-9}  | 100663296x_1x_2^3+1769472x_1x_2^2x_5^2+50331648x_1x_2^2x_5+1458x
     {-3}  | 3x_1^2+8x_1x_2+2x_1x_5-2x_2^3                                   
     ------------------------------------------------------------------------
                                                                             
     _2^7x_5^3-384x_2^7x_5^2+96x_2^6x_5^3-24x_2^5x_5^4+6x_2^4x_5^5+16x_2^2x_5
     8x_5-4608x_2^8+81x_2^7x_5^2+2304x_2^7x_5-864x_2^6x_5^2+216x_2^5x_5^3-54x
     _1x_2x_5^5-10368x_1x_2x_5^4+294912x_1x_2x_5^3+6291456x_1x_2x_5^2-209952x
                                                                             
     ------------------------------------------------------------------------
                                                                             
     ^6+4x_2x_5^7                                                            
     _2^4x_5^4+768x_2^4x_5^3+8192x_2^3x_5^3-144x_2^2x_5^5+4096x_2^2x_5^4-36x_
     _2^9+26244x_2^8x_5+186624x_2^8-2187x_2^7x_5^2-77760x_2^7x_5+221184x_2^7+
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
     2x_5^6+512x_2x_5^5                                                      
     23328x_2^6x_5^2-165888x_2^6x_5-2359296x_2^6-5832x_2^5x_5^3+41472x_2^5x_5
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     ^2+589824x_2^5x_5+25165824x_2^5+1458x_2^4x_5^4-10368x_2^4x_5^3+294912x_2
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     ^4x_5^2+6291456x_2^4x_5+268435456x_2^4+4718592x_2^3x_5^2+201326592x_2^3x
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     _5+3888x_2^2x_5^5-27648x_2^2x_5^4+1966080x_2^2x_5^3+50331648x_2^2x_5^2+
                                                                            
     ------------------------------------------------------------------------
                                                             |
                                                             |
                                                             |
     972x_2x_5^6-6912x_2x_5^5+196608x_2x_5^4+4194304x_2x_5^3 |
                                                             |

             5       1
o7 : Matrix R  <--- R
If noetherNormalization is unable to place the ideal into the desired position after a few tries, the following warning is given:
i8 : R = ZZ/2[a,b];
i9 : I = ideal(a^2*b+a*b^2+1);

o9 : Ideal of R
i10 : (f,J,X) = noetherNormalization I
--warning: no good linear transformation found by noetherNormalization

                                   2       2
o10 = (map(R,R,{a + b, a}), ideal(a b + a*b  + 1), {b})

o10 : Sequence
Here is an example with the option Verbose => true:
i11 : R = QQ[x_1..x_4];
i12 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o12 : Ideal of R
i13 : (f,J,X) = noetherNormalization(I,Verbose => true)
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 20

                                   1     7                        2         
o13 = (map(R,R,{5x  + x  + x , x , -x  + -x  + x , x }), ideal (6x  + x x  +
                  1    2    4   1  2 1   3 2    3   2             1    1 2  
      -----------------------------------------------------------------------
                5 3     73 2 2   7   3     2          2     1 2       7   2
      x x  + 1, -x x  + --x x  + -x x  + 5x x x  + x x x  + -x x x  + -x x x 
       1 4      2 1 2    6 1 2   3 1 2     1 2 3    1 2 3   2 1 2 4   3 1 2 4
      -----------------------------------------------------------------------
      + x x x x  + 1), {x , x })
         1 2 3 4         4   3

o13 : Sequence
The first number in the output above gives the number of linear transformations performed by the routine while attempting to place I into the desired position. The second number tells which BasisElementLimit was used when computing the (partial) Groebner basis. By default, noetherNormalization tries to use a partial Groebner basis. It does this by sequentially computing a Groebner basis with the option BasisElementLimit set to predetermined values. The default values come from the following list:{5,20,40,60,80,infinity}. To set the values manually, use the option LimitList:
i14 : R = QQ[x_1..x_4]; 
i15 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o15 : Ideal of R
i16 : (f,J,X) = noetherNormalization(I,Verbose => true,LimitList => {5,10})
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 10

                1     1             3     10                      3 2   1    
o16 = (map(R,R,{-x  + -x  + x , x , -x  + --x  + x , x }), ideal (-x  + -x x 
                2 1   2 2    4   1  5 1    3 2    3   2           2 1   2 1 2
      -----------------------------------------------------------------------
                   3 3     59 2 2   5   3   1 2       1   2     3 2      
      + x x  + 1, --x x  + --x x  + -x x  + -x x x  + -x x x  + -x x x  +
         1 4      10 1 2   30 1 2   3 1 2   2 1 2 3   2 1 2 3   5 1 2 4  
      -----------------------------------------------------------------------
      10   2
      --x x x  + x x x x  + 1), {x , x })
       3 1 2 4    1 2 3 4         4   3

o16 : Sequence
To limit the randomness of the coefficients, use the option RandomRange. Here is an example where the coefficients of the linear transformation are random integers from -2 to 2:
i17 : R = QQ[x_1..x_4];
i18 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o18 : Ideal of R
i19 : (f,J,X) = noetherNormalization(I,Verbose => true,RandomRange => 2)
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 20
--trying with basis element limit: 40
--trying with basis element limit: 60
--trying with basis element limit: 80
--trying with basis element limit: infinity
--trying random transformation: 2
--trying with basis element limit: 5
--trying with basis element limit: 20

                                                              2          
o19 = (map(R,R,{- 2x  + x , x , 2x  - 3x  + x , x }), ideal (x  - 2x x  +
                    2    4   1    1     2    3   2            1     1 2  
      -----------------------------------------------------------------------
                    2 2       3       2       2           2
      x x  + 1, - 4x x  + 6x x  - 2x x x  + 2x x x  - 3x x x  + x x x x  +
       1 4          1 2     1 2     1 2 3     1 2 4     1 2 4    1 2 3 4  
      -----------------------------------------------------------------------
      1), {x , x })
            4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :